# Approximations of Shannon Mutual Information for Discrete Variables with Applications to Neural Population Coding

## TL;DR

This paper introduces new approximation formulas for Shannon mutual information applicable to discrete variables, demonstrating high accuracy in neural coding contexts and offering practical tools for information theory applications.

## Contribution

The paper develops and validates novel approximation formulas for mutual information that work for both discrete and continuous variables, enhancing computational feasibility in neural coding analysis.

## Key findings

- Approximation formulas are highly accurate for neural population responses.
- One formula performs well regardless of variable discreteness.
- Numerical simulations confirm practical applicability.

## Abstract

Although Shannon mutual information has been widely used, its effective calculation is often difficult for many practical problems, including those in neural population coding. Asymptotic formulas based on Fisher information sometimes provide accurate approximations to the mutual information but this approach is restricted to continuous variables because the calculation of Fisher information requires derivatives with respect to the encoded variables. In this paper, we consider information-theoretic bounds and approximations of the mutual information based on Kullback--Leibler divergence and R\'{e}nyi divergence. We propose several information metrics to approximate Shannon mutual information in the context of neural population coding. While our asymptotic formulas all work for discrete variables, one of them has consistent performance and high accuracy regardless of whether the encoded variables are discrete or continuous. We performed numerical simulations and confirmed that our approximation formulas were highly accurate for approximating the mutual information between the stimuli and the responses of a large neural population. These approximation formulas may potentially bring convenience to the applications of information theory to many practical and theoretical problems.